Sunday, November 20, 2011

You'd be right in saying 'hey, these are just a bunch of overlapping squares.' Yes. The only redeeming thing that I can point to is that they are made by following a rule, and the rule is one that is easy to reproduce without using any external measuring device (like a ruler or protractor), only the squares themselves. Think origami: you find midpoints by folding, etc. In this case, GSP is used, but only simple constructions like mid-point finding and segment creating.

The trick is to find a rule that allows you to start with a square and then construct two points that you can base another square on, and then repeat.

These were made from the same window-pattern instructions mentioned here.

Friday, November 11, 2011

A short while ago I mentioned that A4 paper has nice proportions - it's a silver rectangle, which means that the ratio of its long side to its short side is sqrt(2). Because of their nice proportions, silver rectangles can be used to construct special triangles that we know and love from trigonometry.

One nice way to note the angles in these triangles is to form window patterns based on them - these are shapes made from overlapping pieces of paper that have been rotated according to a rule. The term window pattern comes from William Gibbs - so named because if you put them up in a window, the light shining through the different layers of paper reveals additional patterns and shapes.

Here's one example of the special-triangle-window-pattern process. Start with an A4 or similarly proportioned rectangle, and find the midpoint of one of the shorter sides (by folding the paper, for example).

Now take a second rectangle the same size, and place it so that one vertex lines up with the midpoint drawn, and the other vertex along the same short side of the second rectangle touches the long side of the first. It's easier to see this in a picture:

By doing this, you've constructed the tricky length of sqrt(3)/2 and built the 30-60-90 (pi/6, pi/3, pi/2) triangle. You can confirm that the angle that you've formed a 60 degree triangle by repeating the process and finding that you come "full circle" after 6 pieces of paper (360/6 = 60).

If you change the first placement a bit so that the second rectangle lies mostly across the interior of the first, you get the pattern at the top of the post.

These are nice patterns, but they don't actually use the special properties of A4 (you could do a similar thing with square or letter paper). A little more complicated placing of one rectangle over the other can allow you to create a right triangle with one leg equal to 1 and the other equal to sqrt(2)-1. This is not one of your "standard" special triangles, but it is special in that it allows you to calculate exact values of certain angles (which angles, we'll find out when we complete our pattern).

Here's what the placement looked like that constructed this triangle. I'm afraid that text instructions for the placement would be just too much for this post - maybe you can figure out how it is done from the diagram :).

If you continue placing the rectangles, you will find that it takes 16 of them to come back to the start, which tells us that our triangle contains an angle of pi/8 or 22.5 degrees - the others are pi/2 (90) and 3pi/8 (67.5).

So.. our new special triangle tells us, for example, that tan(pi/8) is equal to sqrt(2)-1 (what other exact values do we get?).

Not much to this post - just playing with the GSP sketch that I pointed to earlier. These are just iterations, plus animation, plus tracing, with what I think are some nice results.

The 'colored Pythagoras tree' fractal below is a classic that I learned in a GSP workshop years ago, and it's based on one of the projects in the free booklet 101 Project Ideas for GSP. I'm sure there are some instructions for the whole thing floating around somewhere. [Update: See the Nov 15th blog post at sine of the times for some instructions on the basic tree.]

The image below is a later stage of the one at the top of the post - an iteration made up of pentagons and curves - the bottom image shows what the first generation of this iteration looks like.

Saturday, November 5, 2011

I found an old GSP file with a bunch geometric fractals in them - I thought that some of them looked nice, so I've posted them here. If you'd like to try them out, you can get the GSP file here - for the most part, they involve pretty standard use of the "iterate" feature.

Animating them in random ways creates some strange looking forms - the same sketch that produces the pentagon fractal above also gives the one below.

The same sketch that gives the snowflake-like pattern at the top of the post gives this odd looking sponge:

Wednesday, November 2, 2011

Because it is my habit to do paper-folding while using public transportation, people sometimes turn their heads and cast pitying eyes upon me; but because of my strong concentration at these times, their looks do not bother me.

- Kazuo Haga, Origamics

The origami model that I fold most frequently is Nick Robinson's A4 butterfly. You can find this model in Nick's book The Origami Bible (unfortunately I don't think that the instructions are posted on his website). Being in North America, A4 paper is not so easily obtained, but luckily I get handed a little piece of almost-the-same-ratio-as-A4-paper every workday morning in the form of a bus transfer.

Rectangles that have the same proportions as A4 paper have nice geometric properties - they aresilver rectangles(named in contrast to golden rectangles), and the niceness of these silver rectangles is due the fact that the ratio of the long side to the short side is sqrt(2). If you don't have an appropriately proportioned bus transfer, or you want to make your own A4-style silver rectangle, Nick Robson provides some helpful instructions here.

Really, you don't need a perfect silver rectangle for the butterfly model - it is pretty forgiving, and tends to work well for bus transfers, ticket stubs, and magazine-subscription inserts (golden rectangles, for example, work too). However, if you look at the simplified crease pattern you can see that the model completely breaks down if the ratio of long to short side is too small or too large. To make things precise, things don't work at all if the ratio of long-side to short-side, r, approaches r = cotan(pi/4) = 1 on the low end, or r = cotan(pi/8) ~= 2.414 on the high end.

The reason that these ratios are as they are is that the fold that creates the outer edge of the wing has an angle of pi/4 with the midline of the paper, while the fold that creates the inner edge of the wing has an angle of pi/8. If either of these lines hit the corner of the rectangle, the model no longer works. That is why the ratio of long-side to short-side (or in trig-ratio speak, adjacent to opposite) is bounded by the cotan of these angles.

Butterflies attempted with almost-square paper have large bodies and almost no wings, while the long paper produces butterflies that have too-long wings and undersized bodies. Although it seems that the model enforces sharply defined boundaries on the range of paper can be used, finding the size of paper that produces the optimal butterfly is another problem. Are silver-rectangle butterflies the best, golden ones, or maybe ones with r = cotan(3pi/16)? This might be a question of personal origami-aesthetics rather than mathematics.